D-MODULES
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திருகு முறுகான பகுப்பிய செயலிகள்
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பெய்லின்சன் -பெர்ன்ஸ்டின் ஒத்திசைவுகள் :
பகுப்பிய செயலிகள் தொகு உரை (ஷீஃப் ) யில் ஓ எக்ஸ் எனும் அல்ஜீப்ரா
உள்ள து. இது பகுப்பிய செயலிகள் தொகு உரையை விவரிப்பது.
எப்படி எனில்
ஓ எக்ஸில் டி எனும் பகுப்பிய செயலிகள் "ஒரே உருமாற்றம்"(ஐஸோமார் ஃ பிசம்) கொண்டது.
இதில் இயல்பான படம் (நேச்சுரல் மேப்) என்பதும் அந்த அல்ஜீப்ராவின் "ஒரே உருமாற்றம்"கொண்டது என்பது அங்கு உள்ளது.
அதாவது கம்மியூட்டேவிடி எனும் "முன் பின் நகர்வு"அதில் உள்ளது. மேற்சொன்ன "தொகு உரை "எனும் ஷீஃப் "திருகு முறுகு "ஆக இருப்பதாக கொள்ளப்படுகிறது.ஏனெனில் அந்த எக்ஸ் மதிப்புக்கோவை "ஒரு போலியான ஒத்தியல்பை "(ஸியூடோ கொஹிரேன்ஸ்") தான் காட்டுகிறது.ஓ எக்ஸ் செயல்
டீ செயலுக்குள்ளும் "உள்ளடங்கி"(இங்குளுசிவ்) இருப்பது ஒரு பொருந்திப் போகாத தன்மையை உண்டாக்கி விடலாம்.\
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டி மோடியூல்களும் "விரிவுரைக்கும் கோட்பாடும்"
(ரெப்ரெசென்ட்டேஷன் தியரி)
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"டி எக்ஸ் "என்பதை எக்ஸ் மீதான ஒரு திருகு முறுக்கு நிறைந்த பகுப்பியல் செயலிகளாகத்தான் காண்கிறோம்.இப்போது இதன் கணிதக்கட்டுமானத்தை கவனிப்போம்.
எல் என்பதை எக்ஸ் கோட்டில் (லைன்) உள்ள "கட்டு"(லைன் பண்டில் ) என்க.
D-MODULES AND REPRESENTATION THEORY
31
By Propositions 2.4 and 2.5, we see that DX is a sheaf of twisted di erential operators on X. This
is an instance of the following more general construction. For L a line bundle on X, de ne a L-twisted
di erential operator on X of order at most n to be a k-linear map d : L
L such that for sections
f0 f1
fn OX, we have
[fn [fn 1 [
[f0 d]]]] = 0
as a k-linear map L L, where a section f OX gives rise to a k-linear map L L via the OX-action
on L. These operators form a sheaf of rings DXL on X, which we call the sheaf of L-twisted di erential
operators on X.17 Notice that a OX-twisted di erential operator on X is simply what we previously
termed a di erential operator on X, and hence that DXOX
= DX. In analogy with DX, we endow DXL
with the order ltration FnDXL. By the following proposition, this construction results in a sheaf of
twisted di erential operators.
Proposition 4.3. For any line bundle L on X, DXL is a sheaf of twisted di erential operators on X.
Proof. We check the conditions in De nition 4.1, though in a di erent order. For (i), an element of
F0DXL is simply an OX-linear map L L, so F0DXL OX because L was a line bundle. For (ii)
and (iii), it is enough to check this locally, in which case LU OU and the statements follow from the
corresponding ones for DU (Propositions 2.4 and 2.5).
By construction, L is a DXL-module, and for L = OX, the DXL-module structure induced on OX
is simply the natural DX-module structure of Example 2.11.
4.2. Lie algebroids and enveloping algebras on the ag variety. Let us now restrict to the case
where X = GB is the ag variety of a connected, simply connected, semisimple algebraic group G. We
now describe a way to understand the notion of a sheaf of g-modules on X. The key advantage of this
notion over simply considering the g-module alone will be the ability to consider all Borel subalgebras
and subgroups of g and G at once.
De ne the Lie algebroid g of the Lie algebra g to be a sheaf of Lie algebras isomorphic to OX k g as
a OX-module and with Lie bracket [
] : g kg gextending the bracket on g such that for f OX
and 1 2 g,wehave
[ 1 f 2]=f[ 1 2]+ ( 1)(f) 2
Then, de ne the universal enveloping algebra U(g) to be the enveloping algebra of g. That is, it is a sheaf
of OX-algebras isomorphic to OX k U(g) as a quasi-coherent OX-module with multiplicative structure
given by extending the structure on U(g) subject to the twisting relation
[
for
f] = ( )(f)
g and f OX. Recall here that : g TX is the map (10) induced by the G-action on X; it
naturally extends to a map of OX-algebras
(14)
: U(g)
DX
For any G-module V, let V := OX k V be the G-equivariant sheaf associated to V as a B-module.
Then, V acquires the structure of a U(g)-module by di erentiating the action of G; explicitly,
g,
viewed as a local section of U(g), acts by
(f
v) = ( )(f) v+f
v
on a local section f v of V. When V is only a U(g)-module, this action still exhibits V as a U(g)
module, though it is no longer a G-equivariant sheaf. Conversely, for any U(g)-module M, its local
sections (UM) acquire the structure of U(g)-modules via the inclusion U(g) , (UOX) kU(g). For
two U(g)-modules M1 and M2, we may equip M1 M2 with the structure of a U(g)-module by letting
a local section
U(g) act via
(m1 m2)= m1 m2+m1 m2
on a local section m1 m2 M1 M2. It is easy to check that this de nes a valid action of U(g) and
that the operation M1
is functorial.
De ne the center Z(g) of U(g) by applying the corresponding construction for Z(g) U(g) and view
Z(g) as a subsheaf of U(g). Further, restricting the map (14) to g U(g), we de ne
b := ker( : g TX)
17If the line bundle L is replaced by a vector bundle M, we may still de ne a sheaf of M-twisted di erential operators.
However, it will not be a sheaf of twisted di erential operators in the sense of De nition 4.1.
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D-MODULES AND REPRESENTATION THEORY
and n := [bb]. Observe that b and n are subsheaves of g. We may describe b explicitly as
b = f g f(x) bx forall x X
where here bx g is the Borel subalgebra corresponding to x X. As a result, we see that
n = f g f(x) nx forall x X
and that bn h := OX k
h. For a map :b OX and a U(g)-module M, we say that U(b) U(g)
acts via
on M if the action of U(b) factors through . If
b
bn h OXit induces also by .18
h is a weight, we denote the map
4.3. A family of twisted di erential operators. Recall from Subsection 3.3 that for each character
e
X(T) we have a G-equivariant line bundle L given by the corresponding one-dimensional
representation of B. Specializing the previous construction, we note that DXL is a sheaf of twisted
di erential operators on X. This construction for DXL requires to be a integral weight (so that
it could lift to a character of T). We now extend it to a family DX of sheaves of twisted di erential
operators parametrized by any weight
h .19
Let
h beanarbitrary weight. We construct a sheaf DX as follows. Consider the map : b OX
given on each ber by taking the action of x on bx. De ne the right ideal I (g) U(g) generated by
the local sections
(
for
)( ) U(g)
b. The following lemma shows that it is possible to quotient by I (g).
Lemma 4.4. For
h an arbitrary weight, I (g) is a two-sided ideal.
Proof. It su ces for us to show that [bg] b. But this follows because the map of (14) was a map of
OX-algebras, hence for any
b and
([
hence [
]
b.
g, we have
]) = ( ) ( ) ( ) ( )=0
Remark. Lemma 4.4 illustrates the essential role that sheaves play in the theory. In particular, a naive
de nition of an analogous left ideal I (g) U(g) generated by
(
)( ) for
in a two-sided ideal. The problem is that b g is not globally the kernel of the map g
considering instead the sheaf b, we are able to detect more local behavior.
Now, de ne the sheaf of OX-algebras DX to be the quotient
DX :=U(g) I (g)
and de ne the maps
(15)
and
(16)
: U(g)
: U(g)
DX
(XDX)
to be the projection map and the map given by composing the action of
b does not result
(XTX). By
on global sections with the
natural inclusion U(g) , (XOX) k U(g), respectively.
Proposition 4.5. For any weight
h , DX is a sheaf of twisted di erential operators.
Proof. We check each property in turn. Condition (i) is obvious. For condition (iii), note that when
restricted to F1U(g) F0U(g)
g, the image of the inclusion I (b)
isomorphism follows from the exact sequence 0
b
g
U(g) is simply b, hence the
TX
0. Finally, for condition (ii), note
that the image of I (g) in FiU(g) Fi 1U(g) Symi
OX
g is given by b Symi
OX
g, hence we see that
gri U(g) I (g) Symi
OX
TX
as needed, where we are using the fact that gb TX.
Corollary 4.6. For any weight
h , U(b) acts via
on DX.
18We refer the reader to [BB93] for more information about constructions of this sort and their generalizations.
19There is a more general theory of twisted di erential operators, for which we refer the reader to [BB93]. However, we
will in this essay only be concerned with sheaves of twisted di erential operators of the form DX.
D-MODULES AND REPRESENTATION THEORY
33
Proof. This follows because the action of U(b) is by applying
Now, let
and the multiplying on the left.
h be an integral weight. Because L is G-equivariant, by Corollary 3.12 we have a
map
: U(g)
OX-algebras
(XDXL ). Together with the inclusion OX
: U(g)
DXL . We would like to show that
DXL , this de nes a map of
realizes DXL
as exactly the
quotient U(g) I (g) of U(g), which would mean that our new family DX extends the family DXL .
Lemma 4.7. Let
h be an integral weight. Then, the map
isomorphism
DX DXL
: U(g)
where here we take corresponding to the choice of B used to construct L .
Proof. First, we claim that the map U(g)
DXL de nes an
DXL kills I (g). For this, it su ces to note that for
each x X, the corresponding Borel subalgebra bx acts on the ber Lx by e x (this is x instead of
because we are on bx instead of b). The resulting map is an isomorphism by the following commutative
diagram on the level of the associated graded
SymOX
TX
grDXL
grDX
where the top and left maps are isomorphisms by Propositions 4.3 and 4.5.
4.4. The localization functor and the localization theorem. Given a DX-module M on X, the
map (16) endows its global sections (XM) with the structure of a U(g)-module. Conversely, given a
U(g)-module M, the map
of (16) allows us to construct the DX-module
Loc (M) := DX U(g)
M
where U(g) is viewed as a (locally) constant sheaf of algebras on X, M is viewed as the corresponding
(locally) constant sheaf of modules over U(g), and the map U(g)
DX is induced by .20 We call
Loc (M) the localization of M and Loc a localization functor. The following shows that it is left adjoint
to the global sections functor := (X ).
Proposition 4.8. The functors Loc : U(g) mod DX mod : are adjoint.
Proof. This follows from the following chain of natural isomorphisms of bifunctors
HomDX
(Loc ( ) )=HomDX
(DX U(g)
)
HomU(g)( HomDX
(DX )) HomU(g)( (X ))
To further characterize the relationship between Loc and , we must analyze
the following theorem, whose proof we defer to Subsection 4.5,
more carefully. By
allows us to identify (XDX) with
U(g) for = .
Theorem 4.9. The map :U(g) (XDX) de nes an isomorphism
U(g) (XDX)
An immediate consequence of Theorem 4.9 is the following.
Corollary 4.10. If M is a U(g)-module for some =
, then Loc (M) = 0.
Proof. Take some z Z(g) with (z)= (z). For any local section
m=
1
(z)
(z)
(z
(z)) m=
1
(z)
mof Loc (M), we have
(z
(z)
where the rst equality follows from Theorem 4.9. We conclude that Loc (M) = 0.
(z)) m =0
20Equivalently, the map factors as U(g)
U(g)
DX.
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D-MODULES AND REPRESENTATION THEORY
Thus, we see that Loc kills the subcategories U(g)
mod of U(g) mod unless = . On the
other hand, by Theorem 4.9, is a functor DX mod U(g)
mod.21 The following theorem and
its corollary, which is the main result of this essay, show that when is regular, these functors give an
equivalence of categories. We defer the proof of Theorem 4.11 to Subsection 4.7.
Theorem 4.11. Let
h be an arbitrary weight. Then, we have the following:
(i) if is dominant, then : DX mod U(g)
(ii) if
is regular, for : DX mod U(g)
mod is exact, and
mod, if (XF) = 0, then F = 0.22
Corollary 4.12 (Beilinson-Bernstein localization). If is regular, then
Loc : U(g)
mod DX mod:
is an equivalence of categories.
Proof. This follows formally from Theorem 4.11. Indeed, by Proposition 4.8, Loc and are adjoint,
hence it su ces to check that the unit and counit maps are isomorphisms.
First, consider the map Loc
id. It becomes an isomorphism after application of by the
adjunction. However, is exact and conservative by Theorem 4.11, which implies that the original map
is an isomorphism. Indeed, for any M U(g)
mod, taking the exact sequence
0
gives an exact sequence
0 (XK)
with (XLoc ( (XM)))
K Loc ( (XM)) M C 0
(XLoc ( (XM))) (XM) (XC) 0
(XM), which shows that (XK) = (XC) = 0, hence C = K = 0.
Next, consider the map id
Loc . For any M U(g)
U(g)I
U(g)J
for M for some index sets I and J. Now, by Theorem 4.11
mod, take a free resolution
M 0
Loc is right exact as the composition of
an exact functor and a left adjoint, hence we obtain the commutative diagram of right exact sequences
U(g)I
U(g)I
U(g)J
U(g)J
M
(XLoc (M))
where the rst two columns are isomorphisms. By the ve lemma, M
phism, completing the proof.
0
0
(XLoc (M)) is an isomor
At rst glance, Corollary 4.12 applies only to weights lying in the principal Weyl chamber
h
i
> 0 . However, because W acts simply transitively on the Weyl chambers, for any weight
which avoids the root hyperplanes
h
i
= 0 , we may nd a unique w W such that
=w( ) is regular. Therefore, Corollaries 3.17 and 4.12 give equivalences of categories
U(g)
mod =U(g)
mod DX mod
Remark. We summarize here the relevant notational conventions underlying the statement of Corollary
4.12. Recall that we chose the positive roots R+ to correspond to the Borel subgroup B, and the G
equivariant line bundle L to correspond to the character e rather than e (we make this choice so
that dominant and regular correspond to globally generated and ample L in Proposition 3.8). In
particular, the b-action on local sections of L is via the weight
rather than . We emphasize,
however, that under our de nition of
DX =U(g)
(
)( ) U(g)
21This fact places Loc in analogy with the following situation for schemes which explains why it is known as the
localization functor. For X a scheme, there is a left adjoint LocX : (XOX) mod QCoh(X) to the functor of global
sections given by LocX(M) := OX
M. For quasicoherent sheaves, LocX is an equivalence if and only if X is a ne;
(XOX)
however, for D-modules, the key consequence of the Beilinson-Bernstein correspondence is that the localization functor is
an equivalence more frequently.
22Recall that a functor F is called conservative if, for F(f) an isomorphism, f is an isomorphism. When F is an exact
functor between abelian categories, the condition of (ii) implies that F is conservative.
D-MODULES AND REPRESENTATION THEORY
35
the sheaf DX corresponds to DXL . In particular, we have that DX DX. As we shall see in the
proof of Proposition 4.13, this shift is necessary to ensure that Z(g) acts by the same central character
on all bers of DX. Finally, we note the action of Z(g) on the global sections of DX-modules is via
(rather than
).
4.5. The map U(g) (XDX). In this subsection, we prove Theorem 4.9 subject to an algebraic
statement related to the geometry of the cone of nilpotent elements in g, which we defer to the next
subsection. We begin by checking that the statement of the theorem makes sense.
Proposition 4.13. The map : U(g) (XDX) factors through U(g) .
Proof. We recall that the map
was obtained by considering
U(g) , (XU(g))
Letting J (g) U(g) be the ideal generated by z
the composition
(XDX)
(z) for z Z(g), it su ces for us to show that
J (g), U(g) DX
is zero. In fact, it su ces to check this on bers. For each x X, by right exactness of kx
that
kx OX
DX =kx OX
DX U(g) (kx OX
I (g))
where we observe that kx OX
I (g) is the right ideal of U(g) generated by
In other words, we have shown that
, we see
( x )( ) for bx.23
kx OX
DX kx U(bx)
U(g)
By Corollary 3.19, the action of Z(g) on this right U(g)-representation is by x
x
= , hence
J (g)x dies on each ber, as needed.
Remark. The proof of Proposition 4.13 allows us to identify
(Xkx OX
DX) kx U(bx)
U(g)
as a right Verma module which under the isomorphism between U(g) and U(g)op will transform to a
dual Verma module.
By Proposition 4.13,
induces a map U(g)
like to show that
(XDX), which we will denote by . We would
is an isomorphism. For this, we will require the following algebraic consequence,
Proposition 4.14, of a theorem of Kostant, Theorem 4.19. Kostants theorem will require some additional
geometric background to state, so for now we describe only Proposition 4.14 and how it may be used to
prove Theorem 4.9. In the next subsection, we will state and prove Kostants theorem and then use it
to derive Proposition 4.14.
Recall now that the G-action on X gives rise to a map : g
the action. This map induces a map Sym(g)
(XTX) of (10) given by di erentiating
(XSymOX
(TX)), which we may characterize as follows.
Proposition 4.14. Let Sym(g)G
+ denote the elements of positive grade in Sym(g)G. The map
Sym(g) Sym(g) Sym(g)G
+ (XSymOX
TX)
is an isomorphism.
Modulo the following lemma, we are now ready to prove Theorem 4.9.
Lemma 4.15. The inclusion Z(g) U(g) gives rise to an isomorphism
grZ(g) Sym(g)G
Proof. By de nition, we have for each i the short exact sequence
0
Fi 1U(g) FiU(g) Symi(g) 0
induced by the order ltration on U(g). Viewing this as a sequence of g-representations, it splits by
complete reducibility. We may therefore apply the functor of G-invariants (which coincides with the
functor of g-invariants) to obtain an exact sequence
0
Fi 1Z(g) FiZ(g) Symi(g)G 0
which induces the desired isomorphism grZ(g) Sym(g)G.
23We emphasize here that I (g)x is not a left ideal of U(g).
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D-MODULES AND REPRESENTATION THEORY
Proof of Theorem 4.9. Equip U(g)
de nition
with a ltration inherited from the ltration of U(g) by order. By
respects the ltration, so it su ces for us to show that the induced map
gr
: grU(g)
gr (XDX)
is an isomorphism. Our approach for this will be to construct a chain of maps
Sym(g) Sym(g) Sym(g)G
+ grU(g)
gr (XDX) , (XgrDX) (XSymOX
TX)
with the rst and last map surjective and injective, respectively, whose composition is the map of
Proposition 4.14. Because it is an isomorphism, each of the maps will be an isomorphism, giving the
claim.
Wenowconstruct the two desired maps. For the rst map, we have a natural surjective map Sym(g)
grU(g) ; it su ces to check that it kills Sym(g)G
+. But recall from Lemma 4.15 that grZ(g)
Sym(g)G, hence for any element f Sym(g)G
+, we see that the image of f in Sym(g) lies in (grZ(g))+
grU(g). But
sends Z(g)
k F0U(g), hence this image dies in grU(g) . For the second map,
we have for all i a short exact sequence of sheaves
0
Fi 1DX FiDX griDX 0
